Derivation of Identric-mean as round-up point for Bias-Free method
Greetings list-members—
…
In 2006, I proposed an allocation divisor-method that I called Bias-Free,
which eliminates bias. I’d like, in this message, to better explain my
derivation of Bias-Free (BF).
…
Instead of defining “bias”, I’ll just let the derivation of BF tell what it
guarantees, and anyone can decide whether that’s unbias.
…
Outline of derivation of Bia-Free (BF):
…
First, to define the terms in the explanation, I should say what a
divisor-method is:
…
Divide total votes by total seats. That’s the Hare Quota.
…
Divide each party’s votes by the Hare Quota, & round off to one of the two
closest integers. (Each divisor-method uses a different round-up point.)
…
Allocate seats according to those rounding-results.
…
If the number of seats thus allocated equals the legally-ordained number of
seats, then that’s the final allocation.
…
Otherwise, try the procedure using another number to replace the Hare
Quota, & call that new number the quota. Repeat the above procedure, using
that new quota instead of the Hare Quota.
…
Find (by trial-&-error, or by some systematic-procedure) a quota such that
the resulting number of seats allocated equals the legally-ordained number
of seats.
…
In the explanation below, “quota” means “quota” as defined above, or a
number of seats equal to the quota. The Hare Quote too is a “quota” as the
term is used below.
…
The object is for the average seats per quota to be unity, averaged over an
interval between two integer numbers of quotas.
…
q = quotas. s = seats. R = the round-up point between a & b.
…
Above the round-up point, s/q = b/q.
…
Below the round-up point, s/q = a/q.
…
…because, below the round-up point a party would have a seats, & above the
round-up point a party would have b seats.
…
Integrate b/q from R, to b.
…
Integrate a/q from a to R.
…
Add the two integrals together.
…
To average over the interval, divide by b – a, the total amount of quota in
the interval.
…
i.e. divide by 1.
…
Set that average s/q in the interval equal to 1, because it’s desired for
it to be 1.
…
Solve for R.
…
R = (1/e)((b^b)/(a^a)).
…
That quantity is called (a special case of) the identric-mean of a & b.
…
Someone expressed concern that the unbias would be spoiled because the size
of parties has a nonuniform probability-distribution. But he didn’t say why
he thinks so, I don’t know what that probability-distribution has to do
with anything said in the derivation.
…
The identric-mean has been much discussed by mathematicians. But,from what
was said in an academic paper (I’ll cite it below), it wasn’t proposed as
the round-up point for an unbiased divisor-method before I proposed it here
in 2006. There were two academic journal-papers about that proposa, in
versions starting in 2008.
…
Here are the two academic-journal references:
…
“The Census and the Second Law: An Entropic Approach to Optimal
Apportionment for the U.S. House of Representatives”.
…
By Andrew E. Charman
…
It was in Physics and Society_, or Journal of Physics and Society, in
2017.
(The latest version of the article was in 2017)
…
The citation said:
…
Cite as arXiv.1712.09440v3 [physics.soc.ph]
…
I don’t know the page or Journal-volume & the issue-numberr, or if that
information is encoded in the numbers above.
…
The other paper was:
…
“Optimal Congressional Apportionment”
…
By Robert A. Agnew.
…
…in The American Mathematical Monthly, for 2008, volume 115, number 4
(April 2008).
…
Pp 297-303 (7 pages)
Just a preliminary remark. Divisor methods and quotas some times
distinguished. Thus there is the Droop quota and corresponding D'Hont
divisor method. Divisor methods regarded as belonging to apportionment,
as by Jefferson and by Webster, not to carve out party seats, which is
too restrictive of personal choice.
On 06/09/2023 05:05, Michael Ossipoff wrote:
Greetings list-members—
…
In 2006, I proposed an allocation divisor-method that I called
Bias-Free, which eliminates bias. I’d like, in this message, to better
explain my derivation of Bias-Free (BF).
…
Instead of defining “bias”, I’ll just let the derivation of BF tell
what it guarantees, and anyone can decide whether that’s unbias.
…
Outline of derivation of Bia-Free (BF):
…
First, to define the terms in the explanation, I should say what a
divisor-method is:
…
Divide total votes by total seats. That’s the Hare Quota.
…
Divide each party’s votes by the Hare Quota, & round off to one of the
two closest integers. (Each divisor-method uses a different round-up
point.)
…
Allocate seats according to those rounding-results.
…
If the number of seats thus allocated equals the legally-ordained
number of seats, then that’s the final allocation.
…
Otherwise, try the procedure using another number to replace the Hare
Quota, & call that new number the quota. Repeat the above procedure,
using that new quota instead of the Hare Quota.
…
Find (by trial-&-error, or by some systematic-procedure) a quota such
that the resulting number of seats allocated equals the
legally-ordained number of seats.
…
In the explanation below, “quota” means “quota” as defined above, or a
number of seats equal to the quota.The Hare Quote too is a “quota” as
the term is used below.
…
The object is for the average seats per quota to be unity, averaged
over an interval between two integer numbers of quotas.
…
q = quotas.s = seats.R = the round-up point between a & b.
…
Above the round-up point, s/q = b/q.
…
Below the round-up point, s/q = a/q.
…
…because, below the round-up point a party would have a seats, & above
the round-up point a party would have b seats.
…
Integrate b/q from R, to b.
…
Integrate a/q from a to R.
…
Add the two integrals together.
…
To average over the interval, divide by b – a, the total amount of
quota in the interval.
…
i.e. divide by 1.
…
Set that average s/q in the interval equal to 1, because it’s desired
for it to be 1.
…
Solve for R.
…
R = (1/e)((b^b)/(a^a)).
…
That quantity is called (a special case of) the identric-mean of a & b.
…
Someone expressed concern that the unbias would be spoiled because the
size of parties has a nonuniform probability-distribution. But he
didn’t say why he thinks so, I don’t know what that
probability-distribution has to do with anything said in the derivation.
…
The identric-mean has been much discussed by mathematicians. But,from
what was said in an academic paper (I’ll cite it below), it wasn’t
proposed as the round-up point for an unbiased divisor-method before I
proposed it here in 2006.There were two academic journal-papers about
that proposa, in versions starting in 2008.
…
Here are the two academic-journal references:
…
“The Census and the Second Law: An Entropic Approach to Optimal
Apportionment for the U.S. House of Representatives”.
…
By Andrew E. Charman
…
It was in Physics and Society_, or Journal of Physics and Society,
in 2017.
(The latest version of the article was in 2017)
…
The citation said:
…
Cite as arXiv.1712.09440v3 [physics.soc.ph http://physics.soc.ph]
…
I don’t know the page or Journal-volume & the issue-numberr, or if
that information is encoded in the numbers above.
…
The other paper was:
…
“Optimal Congressional Apportionment”
…
By Robert A. Agnew.
…
…in The American Mathematical Monthly, for 2008, volume 115, number 4
(April 2008).
…
Pp 297-303(7 pages)
Election-Methods mailing list - seehttps://electorama.com/em for list info
On Tue, Sep 5, 2023 at 23:59 Richard Lung voting@ukscientists.com wrote:
Just a preliminary remark. Divisor methods and quotas some times
distinguished.
I don’t know what you mean by that.
I didn’t say that the divisor methods are quotas. I didn’t say that quotas
are divisor methods.
I defined “quota” for its use in what I was saying. That use of that word
isn’t new or unusual. It’s found in divisor method discussion.
Thus there is the Droop quota and corresponding D'Hont divisor method.
I don’t know what you’ve heard of, but I’ve never heard of Droop quota in a
definition or discussion of d’Hondt.
But maybe someone has proposed a method that he calls “d’Hondt”, & maybe
his method uses the Droop quota, which I’ve heard of being sometimes
proposed, & sometimes used, in STV.
Divisor methods regarded as belonging to apportionment
That term was probably first applied to apportionment proposals, but
nonetheless d’Hondt is the Jefferson divisor method, & Saints-Lague is the
Webster divisor method.
Those two list-PR methods are usually defined, & at least partly
implemented by a systematic procedure, rather than the by the
implementation often or usually specified by the definitions of the divisor
methods proposed & used for apportionment. ..at least in earlier
apportionment discussion.
as by Jefferson and by Webster, not to carve out party seats, which is too
restrictive of personal choice.
Carve out?
I’m not sure, but you seem to be saying that party-list PR restricts
personal choice.
But you didn’t say why you think so.
On 06/09/2023 05:05, Michael Ossipoff wrote:
Greetings list-members—
…
In 2006, I proposed an allocation divisor-method that I called Bias-Free,
which eliminates bias. I’d like, in this message, to better explain my
derivation of Bias-Free (BF).
…
Instead of defining “bias”, I’ll just let the derivation of BF tell what
it guarantees, and anyone can decide whether that’s unbias.
…
Outline of derivation of Bia-Free (BF):
…
First, to define the terms in the explanation, I should say what a
divisor-method is:
…
Divide total votes by total seats. That’s the Hare Quota.
…
Divide each party’s votes by the Hare Quota, & round off to one of the two
closest integers. (Each divisor-method uses a different round-up point.)
…
Allocate seats according to those rounding-results.
…
If the number of seats thus allocated equals the legally-ordained number
of seats, then that’s the final allocation.
…
Otherwise, try the procedure using another number to replace the Hare
Quota, & call that new number the quota. Repeat the above procedure, using
that new quota instead of the Hare Quota.
…
Find (by trial-&-error, or by some systematic-procedure) a quota such that
the resulting number of seats allocated equals the legally-ordained number
of seats.
…
In the explanation below, “quota” means “quota” as defined above, or a
number of seats equal to the quota. The Hare Quote too is a “quota” as
the term is used below.
…
The object is for the average seats per quota to be unity, averaged over
an interval between two integer numbers of quotas.
…
q = quotas. s = seats. R = the round-up point between a & b.
…
Above the round-up point, s/q = b/q.
…
Below the round-up point, s/q = a/q.
…
…because, below the round-up point a party would have a seats, & above the
round-up point a party would have b seats.
…
Integrate b/q from R, to b.
…
Integrate a/q from a to R.
…
Add the two integrals together.
…
To average over the interval, divide by b – a, the total amount of quota
in the interval.
…
i.e. divide by 1.
…
Set that average s/q in the interval equal to 1, because it’s desired for
it to be 1.
…
Solve for R.
…
R = (1/e)((b^b)/(a^a)).
…
That quantity is called (a special case of) the identric-mean of a & b.
…
Someone expressed concern that the unbias would be spoiled because the
size of parties has a nonuniform probability-distribution. But he didn’t
say why he thinks so, I don’t know what that probability-distribution has
to do with anything said in the derivation.
…
The identric-mean has been much discussed by mathematicians. But,from
what was said in an academic paper (I’ll cite it below), it wasn’t proposed
as the round-up point for an unbiased divisor-method before I proposed it
here in 2006. There were two academic journal-papers about that proposa,
in versions starting in 2008.
…
Here are the two academic-journal references:
…
“The Census and the Second Law: An Entropic Approach to Optimal
Apportionment for the U.S. House of Representatives”.
…
By Andrew E. Charman
…
It was in Physics and Society_, or Journal of Physics and Society, in
2017.
(The latest version of the article was in 2017)
…
The citation said:
…
Cite as arXiv.1712.09440v3 [physics.soc.ph]
…
I don’t know the page or Journal-volume & the issue-numberr, or if that
information is encoded in the numbers above.
…
The other paper was:
…
“Optimal Congressional Apportionment”
…
By Robert A. Agnew.
…
…in The American Mathematical Monthly, for 2008, volume 115, number 4
(April 2008).
…
Pp 297-303 (7 pages)
Election-Methods mailing list - see https://electorama.com/em for list info
BTW, Bias-Free is my favorite allocation-rule for list-PR.
But I propose Sainte-Lague, because…
…though SL has a tiny amount of bias in favor of larger parties, it’s so
slight that it’s completely insignificant.
e.g.
150 seats
17 small parties, each with 3% of the vote
… together amounting to 51% of the vote
One big party with 49% of the vote
The 17 small parties each with 3% of the vote together get a majority of
the seats, as they should, with Sainte-Lague.
They can form a majority coalition & a government.
With d’Hondt, the big party gets a big majority.
On Wed, Sep 6, 2023 at 22:04 Michael Ossipoff email9648742@gmail.com
wrote:
On Tue, Sep 5, 2023 at 23:59 Richard Lung voting@ukscientists.com wrote:
Just a preliminary remark. Divisor methods and quotas some times
distinguished.
I don’t know what you mean by that.
I didn’t say that the divisor methods are quotas. I didn’t say that quotas
are divisor methods.
I defined “quota” for its use in what I was saying. That use of that word
isn’t new or unusual. It’s found in divisor method discussion.
Thus there is the Droop quota and corresponding D'Hont divisor method.
I don’t know what you’ve heard of, but I’ve never heard of Droop quota in
a definition or discussion of d’Hondt.
But maybe someone has proposed a method that he calls “d’Hondt”, & maybe
his method uses the Droop quota, which I’ve heard of being sometimes
proposed, & sometimes used, in STV.
Divisor methods regarded as belonging to apportionment
That term was probably first applied to apportionment proposals, but
nonetheless d’Hondt is the Jefferson divisor method, & Saints-Lague is the
Webster divisor method.
Those two list-PR methods are usually defined, & at least partly
implemented by a systematic procedure, rather than the by the
implementation often or usually specified by the definitions of the divisor
methods proposed & used for apportionment. ..at least in earlier
apportionment discussion.
as by Jefferson and by Webster, not to carve out party seats, which is too
restrictive of personal choice.
Carve out?
I’m not sure, but you seem to be saying that party-list PR restricts
personal choice.
But you didn’t say why you think so.
On 06/09/2023 05:05, Michael Ossipoff wrote:
Greetings list-members—
…
In 2006, I proposed an allocation divisor-method that I called Bias-Free,
which eliminates bias. I’d like, in this message, to better explain my
derivation of Bias-Free (BF).
…
Instead of defining “bias”, I’ll just let the derivation of BF tell what
it guarantees, and anyone can decide whether that’s unbias.
…
Outline of derivation of Bia-Free (BF):
…
First, to define the terms in the explanation, I should say what a
divisor-method is:
…
Divide total votes by total seats. That’s the Hare Quota.
…
Divide each party’s votes by the Hare Quota, & round off to one of the
two closest integers. (Each divisor-method uses a different round-up point.)
…
Allocate seats according to those rounding-results.
…
If the number of seats thus allocated equals the legally-ordained number
of seats, then that’s the final allocation.
…
Otherwise, try the procedure using another number to replace the Hare
Quota, & call that new number the quota. Repeat the above procedure, using
that new quota instead of the Hare Quota.
…
Find (by trial-&-error, or by some systematic-procedure) a quota such
that the resulting number of seats allocated equals the legally-ordained
number of seats.
…
In the explanation below, “quota” means “quota” as defined above, or a
number of seats equal to the quota. The Hare Quote too is a “quota” as
the term is used below.
…
The object is for the average seats per quota to be unity, averaged over
an interval between two integer numbers of quotas.
…
q = quotas. s = seats. R = the round-up point between a & b.
…
Above the round-up point, s/q = b/q.
…
Below the round-up point, s/q = a/q.
…
…because, below the round-up point a party would have a seats, & above
the round-up point a party would have b seats.
…
Integrate b/q from R, to b.
…
Integrate a/q from a to R.
…
Add the two integrals together.
…
To average over the interval, divide by b – a, the total amount of quota
in the interval.
…
i.e. divide by 1.
…
Set that average s/q in the interval equal to 1, because it’s desired for
it to be 1.
…
Solve for R.
…
R = (1/e)((b^b)/(a^a)).
…
That quantity is called (a special case of) the identric-mean of a & b.
…
Someone expressed concern that the unbias would be spoiled because the
size of parties has a nonuniform probability-distribution. But he didn’t
say why he thinks so, I don’t know what that probability-distribution has
to do with anything said in the derivation.
…
The identric-mean has been much discussed by mathematicians. But,from
what was said in an academic paper (I’ll cite it below), it wasn’t proposed
as the round-up point for an unbiased divisor-method before I proposed it
here in 2006. There were two academic journal-papers about that
proposa, in versions starting in 2008.
…
Here are the two academic-journal references:
…
“The Census and the Second Law: An Entropic Approach to Optimal
Apportionment for the U.S. House of Representatives”.
…
By Andrew E. Charman
…
It was in Physics and Society_, or Journal of Physics and Society, in
2017.
(The latest version of the article was in 2017)
…
The citation said:
…
Cite as arXiv.1712.09440v3 [physics.soc.ph]
…
I don’t know the page or Journal-volume & the issue-numberr, or if that
information is encoded in the numbers above.
…
The other paper was:
…
“Optimal Congressional Apportionment”
…
By Robert A. Agnew.
…
…in The American Mathematical Monthly, for 2008, volume 115, number 4
(April 2008).
…
Pp 297-303 (7 pages)
Election-Methods mailing list - see https://electorama.com/em for list info
When I happened to remark that divisor methods and quota methods are
sometimes distinguished, this is so. Robert Newland did so, in his book
on Comparitive Electoral Systems.
Indeed, I didn't say why party lists restrict personal choice. (And you
didn't deny it.) But Enid Lakeman did say why, in How Democracies Vote.
In so many words, It is because with transferable voting, the voters
decide how their votes will count from first preferences onward, while
with party lists, it is the parties who more or less determine personal
representation.
Richard Lung.
On 07/09/2023 06:04, Michael Ossipoff wrote:
On Tue, Sep 5, 2023 at 23:59 Richard Lung voting@ukscientists.com wrote:
Just a preliminary remark. Divisor methods and quotas some times
distinguished.
I don’t know what you mean by that.
I didn’t say that the divisor methods are quotas. I didn’t say that
quotas are divisor methods.
I defined “quota” for its use in what I was saying. That use of that
word isn’t new or unusual. It’s found in divisor method discussion.
Thus there is the Droop quota and corresponding D'Hont divisor
method.
I don’t know what you’ve heard of, but I’ve never heard of Droop quota
in a definition or discussion of d’Hondt.
But maybe someone has proposed a method that he calls “d’Hondt”, &
maybe his method uses the Droop quota, which I’ve heard of being
sometimes proposed, & sometimes used, in STV.
Divisor methods regarded as belonging to apportionment
That term was probably first applied to apportionment proposals, but
nonetheless d’Hondt is the Jefferson divisor method, & Saints-Lague is
the Webster divisor method.
Those two list-PR methods are usually defined, & at least partly
implemented by a systematic procedure, rather than the by the
implementation often or usually specified by the definitions of the
divisor methods proposed & used for apportionment. ..at least in
earlier apportionment discussion.
as by Jefferson and by Webster, not to carve out party seats,
which is too restrictive of personal choice.
Carve out?
I’m not sure, but you seem to be saying that party-list PR restricts
personal choice.
But you didn’t say why you think so.
On 06/09/2023 05:05, Michael Ossipoff wrote:
Greetings list-members—
…
In 2006, I proposed an allocation divisor-method that I called
Bias-Free, which eliminates bias. I’d like, in this message, to
better explain my derivation of Bias-Free (BF).
…
Instead of defining “bias”, I’ll just let the derivation of BF
tell what it guarantees, and anyone can decide whether that’s unbias.
…
Outline of derivation of Bia-Free (BF):
…
First, to define the terms in the explanation, I should say what
a divisor-method is:
…
Divide total votes by total seats. That’s the Hare Quota.
…
Divide each party’s votes by the Hare Quota, & round off to one
of the two closest integers. (Each divisor-method uses a
different round-up point.)
…
Allocate seats according to those rounding-results.
…
If the number of seats thus allocated equals the legally-ordained
number of seats, then that’s the final allocation.
…
Otherwise, try the procedure using another number to replace the
Hare Quota, & call that new number the quota. Repeat the above
procedure, using that new quota instead of the Hare Quota.
…
Find (by trial-&-error, or by some systematic-procedure) a quota
such that the resulting number of seats allocated equals the
legally-ordained number of seats.
…
In the explanation below, “quota” means “quota” as defined above,
or a number of seats equal to the quota.The Hare Quote too is a
“quota” as the term is used below.
…
The object is for the average seats per quota to be unity,
averaged over an interval between two integer numbers of quotas.
…
q = quotas.s = seats.R = the round-up point between a & b.
…
Above the round-up point, s/q = b/q.
…
Below the round-up point, s/q = a/q.
…
…because, below the round-up point a party would have a seats, &
above the round-up point a party would have b seats.
…
Integrate b/q from R, to b.
…
Integrate a/q from a to R.
…
Add the two integrals together.
…
To average over the interval, divide by b – a, the total amount
of quota in the interval.
…
i.e. divide by 1.
…
Set that average s/q in the interval equal to 1, because it’s
desired for it to be 1.
…
Solve for R.
…
R = (1/e)((b^b)/(a^a)).
…
That quantity is called (a special case of) the identric-mean of
a & b.
…
Someone expressed concern that the unbias would be spoiled
because the size of parties has a nonuniform
probability-distribution. But he didn’t say why he thinks so, I
don’t know what that probability-distribution has to do with
anything said in the derivation.
…
The identric-mean has been much discussed by mathematicians.
But,from what was said in an academic paper (I’ll cite it below),
it wasn’t proposed as the round-up point for an unbiased
divisor-method before I proposed it here in 2006.There were two
academic journal-papers about that proposa, in versions starting
in 2008.
…
Here are the two academic-journal references:
…
“The Census and the Second Law: An Entropic Approach to Optimal
Apportionment for the U.S. House of Representatives”.
…
By Andrew E. Charman
…
It was in _Physics and Society__, or _Journal of Physics and
Society_, in 2017.
(The latest version of the article was in 2017)
…
The citation said:
…
Cite as arXiv.1712.09440v3 [physics.soc.ph <http://physics.soc.ph>]
…
I don’t know the page or Journal-volume & the issue-numberr, or
if that information is encoded in the numbers above.
…
The other paper was:
…
“Optimal Congressional Apportionment”
…
By Robert A. Agnew.
…
…in The American Mathematical Monthly, for 2008, volume 115,
number 4 (April 2008).
…
Pp 297-303(7 pages)
----
Election-Methods mailing list - seehttps://electorama.com/em for list info